NCTM Feedback on Problem Solving 2.pptx
4/18/15 Getting Students Invested in the Process of Problem Solving Debbie Wile [email protected] Annie Fe8er [email protected] h8p://mathforum.org/boston/ A Brief History… • Started as a Elementary Language Arts teacher in private school for bright kids with reading disabiliHes. • Moved to public school in 1990. • Introduced to Math Forum and Problems of the Week in mid 1990s. 1 4/18/15 Digression: Introducing PoWs… Mathforum.org 2 4/18/15 3 4/18/15 4 4/18/15 Why I use PoWs in the first place… Math belongs in the real world InteresHng and varied problems PracHce wriHng about math Align with State test prep Use them with grades 1-‐5 but with different goals • Same problem with kids of different ages • • • • • Why I use PoWs in the first place… • Kids love using the computer! • Computer plaVorm allows for dialogue between teacher and student(s) • Focus on the PROCESS • FUN!!! 5 4/18/15 Limitations of PoW…. • Need computer access • They take !me!!! – “Grading” • Like mini-‐essays – During class • Away from curriculum • Totally worthwhile How a Lesson Works… • Many permutaHons over the years – Need to teach the soaware (less now) – Submit in class vs at home – Work in teams vs independently – How much “hand holding” to do – Use the problems off-‐line • Scheduling limitaHons – InserHng into regular curriculum – Push-‐in model 6 4/18/15 Recent Lesson Structure… In class: – Always start with scenario – NoHcing and wondering – 3-‐5 minutes of silent thinking with paper – Share and conHnue working with peers – Teacher led mini-‐lessons as needed with groups or whole class – InvitaHon to lunchHme assistance Recent Lesson Homework… • At home: – Write your submission 7 4/18/15 Introducing… EnCoMPASS!!! • …focused on understanding and improving mathemaHcal thinking through work with formaHve assessment rubrics and feedback to student problem solving…. • Summer insHtute in 2013 • Look at how to maximize impact of feedback from teachers • EnCoMPASS soaware (currently in development) h8p://mathforum.org/encompass 8 4/18/15 The Question… How can I maximize the impact of wri8en feedback given to students? The Research… • PA State teacher evaluaHon system based on Charlo8e Danielson – Two year cycle for acHon research • EnCoMPASS grant – ExperHse and personnel from Swarthmore College – Classroom observers – Interviews three Hmes per year for two years – De-‐idenHfied data connected to interviews 9 4/18/15 The EnCoMPASS software concept… Aaer looking at a student’s work, point out two parts of the student’s work: 1. A part that represents a strength 2. A part that needs more work Feathers and Fur [Problem #421] Xiao, who lives in New York, is a lover of homing pigeons. She also trains seeing-‐eye dogs. When I spoke to her recently, she told me in her quirky way that she was currently hosHng 36 heads and 80 feet. I was lea to figure out how many puppies and pigeons she had. Can you help me? 10 4/18/15 11 4/18/15 Results… 1. Focus on answer genng 2. Student frustraHon and work avoidance 3. Parents did the work 4. I spent a lot of Hme wriHng comments that didn’t seem to help the students. 12 4/18/15 Brick Wall Effect… Dear Ms.Wile, I don't really under stand the extra problem I under stand the regular problem absouloutly fine but the extra not so much. My tried to explain it to me but when I tried this is what I got. I have a feeling that this is incorrect. Just wanted you to let you know and if next Hme I see you ,you can talk to me about it. Thank you! A’s Moons by Jove, 3-‐26-‐14 K’s Moons by Jove, 3-‐26-‐14 Once again, when I looked over the numbers, there weren't ANY common mulHples of all three numbers I was looking for, (which are seven, twelve, and sixteen, in case you've forgo8en). Now, I felt really discouraged! How could I find the informaHon I needed without spending three hours to get it? When I asked my , he had the perfect soluHon. "You should use Excel", he told me. I started to feel very silly. Why hadn't I thought of that? 13 4/18/15 Stop! In the name of math… • April • Time to change gears! • Something radically different… Analyze the problem… • Focus on answer • Assume students can persevere independently • ReflecHon is devalued 14 4/18/15 A New Approach… Focus on process – Want the kids to pay a8enHon to their thinking – Change goal from answer genng to sharing your thinking – Make it a conversaHon – Interact with the teacher The first, NEW class… 1. Read problem 2. Stop 3. Write your emoHonal response to problem (give examples) 4. Stand up -‐-‐ turn around clockwise 5. “What you just wrote is reflecHon” (label as such on your paper) 6. 5 minutes of independent work on the problem 7. Write what you just did and why 8. stand up -‐-‐ turn around counter clockwise 9. “What you just wrote is explanaHon of your thinking” 10. repeat steps 3-‐9 15 4/18/15 The first, NEW homework… Take what you wrote down and type this into submission. THIS IS WHAT WE WANT: A COMBINATION OF YOUR PERSONAL REACTIONS AND YOUR THINKING ABOUT PROBLEM! 1. Spend no more than 15 minutes on it. 2. I will read it the next day and respond. 3. You then read my comments and conHnue working. 4. Repeat steps 1-‐3 unHl “You’re done!” 5. Parents prohibited! Teresa's Tiles [Problem #2803] Teresa is going to put down new ceramic tiles on her bathroom floor. She has selected square tiles that are 4 inches on each side. These are the kind of tiles that can be placed right next to each other without leaving additional space for grout. At The Home Station, she learned how to cut the tiles in case she needs any fractional pieces to cover her floor completely. This diagram of the bathroom floor shows the dimensions of the floor space she needs to cover. The sink area does not get tiled. Questions: How many tiles will she need to buy to cover her floor? How many tiles will she have to cut in order to cover the entire space? 16 4/18/15 Teresa's Tiles [Problem #2803] Teresa is going to put down new ceramic tiles on her bathroom floor. She has selected square tiles that are 4 inches on each side. These are the kind of tiles that can be placed right next to each other without leaving additional space for grout. At The Home Station, she learned how to cut the tiles in case she needs any fractional pieces to cover her floor completely. This diagram of the bathroom floor shows the dimensions of the floor space she needs to cover. The sink area does not get tiled. 18”x32” 60”-‐28”=32” Questions: How many tiles will she need to buy to cover her floor? How many tiles will she have to cut in order to cover the entire space? (She needs 189 Hles and 8 must be cut!) NEW Results … • Perseverance increases dramaHcally • Asks quesHons • I can answer them and give guidance without doing the work • Focus on problem solving process, NOT on the answer • Parents stopped doing the work 17 4/18/15 A’s Theresa’s Tiles, 4-‐1-‐14 What I decided to do was re-‐ draw the diagram of the bathroom in my math notebook. I thought that maybe I should add the numbers of the perimeters that are on each side of the bathroom before I did anything else. When I added them together I got 190 inches. I then thought that maybe I needed to figure out the the perimeter that is around the sink. But is says in the problem that the bathroom area does not get Hled. So that means that I don't need to do that. I now need to figure out how much Hles will fit into the area of the bathroom. My response 4-‐2-‐14 I wonder if you can easily explain the difference between perimeter and area. Which one of these is important to this problem? 18 4/18/15 A’s second submission TT 4-‐2-‐14 What I did before Wednesday class: I realized this is all wrong … Area is important to this problem because you need to figure out how many Hles are going onto the floor if it was perimeter it would only be the edges of the bathroom area would be the filling of the bathrooms or the Hling. Dear ms.wile, I did what my friend did. Is that OK? I completely understand what she did. I don't understand how to get the sink perimeter though. First I drew a picture using 4 pieces of graph paper. I traced 60 squares on the top, 60 squares on the side, 42 squares on the bo8om, big blanked traced area on the lower lea, and 28 squares on the upper lea. That represents all the inches of each side. My response 4-‐4-‐14 You are perfectly fine modeling your thinking aaer your friend, especially since it makes sense to you. I wonder if you're trying to figure out the area or the perimeter of the bathroom? I think that's where you're genng stuck. Keep trying and I'll read what you write next. 19 4/18/15 A’s TT 4-‐2-‐14 of each side. That represents all the inches A’s TT End 4-‐10-‐14 Then, I mulHplied 60 by 28, because the top perimeter is 60 and the right half side is 28. I got 600. I then added 19 to 600 because the 1 part sink corner perimeter is 19 Hles. That total equals 619. Another early example… • My feedback – Points out where further explanaHon is needed – Points out calculaHon or reasoning errors • Check out the reflecHon! 20 4/18/15 J’s TT 4-‐2-‐14 I thought that since we were going to break it down step by step it would be easier, but I was wrong. I thought that it would take longer to do the homework. It doesn't really make much sense to me. Hard problem, but homework that doesn't take that long. The answer is that 136 Hles are needed, and 8 Hles are cut My response 4-‐2-‐14 No, the number 136 is not correct. Maybe if you explain where you got that number, I can help you. Figuring out the dimensions of the sink might help you, but I'm not sure what you're thinking. 21 4/18/15 J’s second TT 4-‐3-‐14 Next I tried to find a number that best fits into all of those numbers. the best number I could find is 4 Next I divided 60 into 4 that = 15. So I got out a piece of graph paper and drew a square with 15 boxes on each side. Now I divided 32 into 4 which = 8, and 18 into 4 which = 4.5 So in my square I went from the lea corner where the sink is 4.5 boxes over, then I went up 8 boxes from the lea corner. I put my 2 fingers there unHl they met up. Now I have my sink drawn. I worked my way from the right side 10 spaces to the lea. since it is 15 boxes down and 10 boxes to the side the total number of boxes should be 150. Now I go from the lea side to the right side 4 spaces. Since there are only 7 boxes down because of the sink. So I should have 28. I added 150 + 28 = 178. Now I have to do the last box. There are 7 full Hles. There are 8 half . cut Hles. So I count 2 half cut flies as 1 Hle and get 4. 7 + 4 = 11. So now I add 178 + 11 = 189. But since each box = 4 Hles, I have to mulHply 189 x 4 = 756. Now I have to mulHply how many Hles were cut. 8 x 4 = 32 So my answer is 756 Hles, and 32 Hles were cut My response 4-‐4-‐14 You are so very close!!!! Your explanaHon is very clear and complete. I can see exactly where your thinking is wrong, so I can show you. At the very end you wrote, " But since each box = 4 Hles, I have to mulHply 189 x 4 = 756." Think about that again. Each box = 4 INCHES (not Hles) and each Hle is how big? 22 4/18/15 J’s last TT 4-‐7-‐14 Now I have to do the last box. There are 7 full Hles. There are 8 half . cut Hles. So I count 2 half cut flies as 1 Hle and get 4. 7 + 4 = 11. So now I add 178 + 11 = 189. So my answer is 189 Hles, and 8 Hles were cut. reflec!on I really thought this problem was going to be impossible. I literally could have crawled up in a ball and say. "Hide me from my Ms. Wile problem". But once [a kid] showed the class the way he did the problem, my brain turned on and went to work. I have to do another pow because I have completed 1 so far. I think that if You give us one that is not really difficult, it would give all of us a break and a chance to get one more pow done. Year 2: Application… • • • • Start by focusing on the process Baby steps first Gradual release of responsibility It’s all worth the effort! 23 4/18/15 Review: Classroom procedure… 1. Read problem 2. Stop 3. Write your emoHonal response to problem (give examples) 4. Stand up -‐-‐ turn around clockwise 5. “What you just wrote is reflecHon” (label as such on your paper) 6. 5 minutes of independent work on the problem 7. Write what you just did and why 8. stand up -‐-‐ turn around counter clockwise 9. “What you just wrote is explanaHon of your thinking” 10. repeat steps 3-‐9 Review: Homework… Take what you wrote down and type this into submission. THIS IS WHAT WE WANT: A COMBINATION OF YOUR PERSONAL REACTIONS AND YOUR THINKING ABOUT PROBLEM! 1. Spend no more than 15 minutes on it. 2. I will read it in the morning and respond. 3. You can read my comments and conHnue working. 4. Repeat steps 1-‐3 unHl “You’re done!” 5. Parents prohibited 24 4/18/15 Super Deal Salsa [Problem #829] Perusing the super deals aisle at the grocery store the other day, I found some salsa. Since Frank loves chips and salsa, I decided to purchase two tubs. Each tub cost $3.49 and held 4 pounds 6 ounces of salsa. We usually buy 20-‐ounce jars for $2.29. How much money did I save by purchasing the super deal salsa? Extra: Give the unit cost for each of the salsas. Super Deal Salsa [Problem #829] Perusing the super deals aisle at the grocery store the other day, I found some salsa. Since Frank loves chips and salsa, I decided to purchase two tubs. Each tub cost $3.49 and held 4 pounds 6 ounces of salsa. We usually buy 20-‐ounce jars for $2.29. How much money did I save by purchasing the super deal salsa? Extra: Give the unit cost for each of the salsas. 2 tubs @ $3.49 = $6.98 for 140 oz. 7 jars @ $2.29 = $16.03 for 140 oz. Save $9.05 25 4/18/15 Ch’s #1 10-‐2-‐14 I'm thinking that they would save about one dollar on buying the tubs of salsa. I know that 16oz=1lb. So 20oz=1lb and 4oz. 1 of the 20oz jars contains 1lb and 4oz of salsa. I was wondering if there was a more or less expensive deal if chips were bought in like a bundle. That's what I was thinking about this problem. My first response 10-‐3-‐14 Hi Ch, I love your thinking about the markeHng end of this problem, with a chips/salsa package deal. When we finish the problem, you could write an "Extra" about this and I'll get it to the people at Math Forum. Who knows, you might get it published. You wrote, "I know that 16oz=1lb. So 20oz=1lb and 4oz. 1 of the 20oz jars contains 1lb and 4oz of salsa." This is completely correct and you're on the right track to convert so both tubs and jars are in similar units. Why don't you go ahead and try a next step and write to me about it. I can help you from there. Ms. Wile 26 4/18/15 Ch’s #2 10-‐7-‐14 Solving The Problem Since 16 oz=1lb 20oz=1lb and 4oz. 2 20oz jars contain 2lb and 8 ounces. The price for that would be $4.58. The tub already has more than 2 20oz jars, now let's try 3 20oz jars. 3 jars would contain 3lb and 12oz. A tub already has more than 3 jars. Now we'll try 4 20oz jars. 4 20oz jars contain 5lbs. 5lbs is more than 4lb and 6oz. Two tubs would contain 8lb and 12oz. That's the first step for me into solving this problem. P. S What does it mean by the unit cost? My second response 10-‐8-‐14 Hi Ch, Everything you've done so far is totally correct and easy to understand. I like how you keep checking to compare tubs and jars each Hme you add another jar. You wrote, "Two tubs would contain 8lb and 12oz." Correct. Now that you know that, you can compare that to a certain number of jars. How many jars? Then you can figure out the price difference. Once you've go8en this done, I'll help you with unit cost. Ms. Wile 27 4/18/15 Ch’s #3 on 10-‐13-‐14 Solving The Problem Since 16 oz=1lb 20oz=1lb and 4oz. 2 20oz jars contain 2lb and 8 ounces. The price for that would be $4.58. … My response 10-‐13-‐14 Ch, Your thinking is excellent and you're almost done! You wrote, "Two tubs would cost $6.98. Seven jars would cost $16.03. She saved about $9.11." Please check your last calculaHon because your subtracHon is incorrect. Also, please re-‐read everything you wrote and think about the organizaHon of your ideas. How could you move your sentences around to make more sense? Ms. Wile 28 4/18/15 Ch’s #4 on 10-‐14-‐14 …Two tubs would cost $6.98. Seven jars would cost $16.03. $16.03-‐$6.98=$9.05. She saved about $9.05. I really don't know what I need to change sentence-‐wise. Ch’s #4 on 10-‐15-‐14 Here's my extra: I just remembered that we are running low on chips. So I've kept looking down the isles and saw a "Super bundle". The super bundle had a bag of chips and a tub of salsa. The whole bundle cost $5.24. A regular bag of chips cost 4.78. We usually buy a bag and a tub of salsa. How much money did we save on the "super bundle"? 29 4/18/15 Even More Results… • Students engaged • Teacher overwhelmed • Do what I say: It’s w orth the e ffort… really! (And it’s actually F-‐U-‐N to see PERSEVERE! them grow.) 30 4/18/15 31 4/18/15 Remember the Research? Preliminary data analysis -‐-‐ Three types of students: 1. Student 1 has an interest in mathemaHcs 2. Student 2 had less interest in mathemaHcs at the beginning of the year. Student 2 conHnues to be less interested. 3. Student 3 also had less interest in mathemaHcs at the beginning of the year; however, Student 3 appears to be developing an interest. Questions for Further Research… • How to customize type and frequency of feedback to the three types of students • How to gradually diminish the amount of feedback most efficiently • ??? 32 4/18/15 Thanks for coming! h8p://mathforum.org/boston/ 33
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